nLab
Buchstaber–Rees theorem

under construction

Idea

Gelfand and Kolmogorov proved in 1939 that a compact Hausdorff topological space XX can be canonically embedded into the infinite‐dimensional vector space C(X) *C(X)^*, the dual space of the algebra of continuous functions C(X)C(X), as an “algebraic variety”, specified by an infinite system of quadratic equations.

Buchstaber and Rees have extended this to all symmetric powers Sym n(X)Sym_n(X) using their notion of the Frobenius nn‐homomorphisms.

References

  • Victor M. Buchstaber, Elmer G. Rees, The Gelfand map and symmetric products, Selecta Math. 8 (2002), no. 4, 523–535; …, Russian Math. Surveys 59 (1(355)):125–145, 2004; Frobenius n-homomorphisms, transfers and branched coverings, Math. Proc. Cambr. Phil. Soc. 2007, arXiv:math.RA/0608120 .

  • I. M. Gel’fand, A. N. Kolmogorov. Dokl. Akad. Nauk SSSR 22: 11–15, 1939.

  • H. M. Khudaverdian, Th. Th. Voronov, dadada, Lett. Math. Phys., 74(2):201–228, 2005; Operators on superspaces and generalizations of the Gelfand–Kolmogorov theorem, AIP Conf. Proc. 956, 149 (2007);doi; On generalized symmetric powers and a generalization of Gel’fand-Kolmogorov-Buchstaber-Rees theorem, pdf; A short proof of the Buchstaber–Rees theorem, Phil. Trans. R. Soc. A 369, no. 1939, 1334-1345 (2011) full, pdf (keywords: Berezinian (superdeterminant), Frobenius recursion, symmetric powers, maps of algebras, nn-homomorphism, p|qp|q-homomorphism

  • Alexander Pavlov, Evgenij Troitsky, Quantization of branched coverings, Russian J. of Math. Phys. 18, N 3 (2011), 338-352 arxiv/1002.3491

Last revised on August 13, 2018 at 06:29:39. See the history of this page for a list of all contributions to it.